RC24306 (W0707-086) July 11, 2007Mathematics

IBM Research Report

Applications of OR to Finance

Aliza R. Heching, Alan J. KingIBM Research Division

Thomas J. Watson Research CenterP.O. Box 218

Yorktown Heights, NY 10598

Research DivisionAlmaden - Austin - Beijing - Haifa - India - T. J. Watson - Tokyo - Zurich

LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a ResearchReport for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specificrequests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T. J. Watson Research Center , P.O. Box 218, Yorktown Heights, NY 10598 USA (email: reports@us.ibm.com). Some reports are available on the internet at http://domino.watson.ibm.com/library/CyberDig.nsf/home .

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1. Introduction

Operations Research provides a rich set of tools and techniques that are applied to

financial decision making. The first topic that likely comes to mind for most readers is

Markowitz’s Nobel Prize winning treatment of the problem of portfolio diversification

using quadratic programming techniques. This treatment, which first appeared in 1952,

underlies almost all of the subsequent research into the pricing of risk in financial

markets. Linear programming, of course, has been applied in many financial planning

problems, from the management of working capital to formulating a bid for the

underwriting of a bond issue. Less well known is the fundamental role that duality theory

plays in the theoretical treatment of the pricing of options and contingent claims, both in

its discrete state and time formulation using linear programming and in its continuous

time counterparts. This duality leads directly to the Monte Carlo simulation method for

pricing and evaluating the risk of options portfolios for investment banks; this activity

probably comprises the single greatest use of computing resources in any industry.

This chapter does not cover every possible topic in the applications of Operations

Research (“OR”) to Finance. We have chosen to highlight the main topics in investment

theory and to give an elementary, mostly self-contained, exposition of each. A

comprehensive perspective of the application of OR techniques to financial markets

along with an excellent bibliography of the recent literature in this area can be found in

the survey by Board et al. (2003). In this chapter we chose not to cover the more

traditional applications of OR to financial management for firms, such as the

management of working capital, capital investment, taxation, and financial planning. For

these, we direct the reader to consult Ashford et al. (1988). We also excluded financial

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forecasting models; the reader may refer to Campbell et al. (1997) and Mills (1999) for

recent treatments of these topics. Finally, Board et al. (2003) provide a survey of the

application of OR techniques for the allocation of investment budgets between a set of

projects. Complete and up-to-date coverage of finance and financial engineering topics

for readers in Operations Research and Management Science may be found in the

handbooks of Jarrow, Maksimovic and Ziemba (1995) and the forthcoming volume of

Birge and Linetsky (forthcoming, 2007).

We begin this chapter by introducing some basic concepts in investment theory.

In Section 2 we present the formulas for computing the return and variance of return on a

portfolio. The formulas for a portfolio’s mean and variance presume that these

parameters are known for the individual assets in the portfolio. In Section 3 we discuss

two methods for estimating these parameters when they are not known.

Section 4 explains how a portfolio’s overall risk can be reduced by including a

diverse set of assets in the portfolio. In Section 5 we introduce the risk-reward tradeoff

efficient frontier and the Markowitz problem. Up to this point, we have assumed that the

investor is able to specify a mathematical function describing his attitude toward risk. In

Section 6, we consider utility theory which does not require an explicit specification of a

risk function. Instead, utility theory assumes that investors specify a utility, or

satisfaction, with any cash payout. The associated optimal portfolio selection problem

will seek to maximize the investor’s expected utility.

Section 7 discusses the Black-Litterman model for asset allocation. Black and

Litterman use Bayesian updating to combine historical asset returns with individual

investor views to determine a posterior distribution on asset returns which is used to

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make asset allocation decisions. Section 8 considers the challenges of risk management.

We introduce the notion of coherent risk measures and conditional value-at-risk (CVaR),

and show how a portfolio selection problem with a constraint on CVaR can be formulated

as a stochastic program.

In Sections 9 through 13 we turn to the problem of options valuation. Options

valuation combines a mathematical model for the behavior of the underlying uncertain

market factors with simulation or dynamic programming (or combinations thereof) to

determine options prices. Section 14 considers the problem of asset-liability matching in

a multi-period setting. The solution uses stochastic optimization based upon Monte Carlo

simulation. Finally, in Section 15 we present some concluding remarks.

2. Return

Suppose that an investor invests in asset i at time 0 and sells the asset at time t.

The rate of return (more simply referred to as the return) on asset i over time period t is

given by:

00

timeatiassetininvestedamounttimeatiassetininvestedamountttimeatreceivedamountri

−= . (1)

Now suppose that an investor invests in a portfolio of N assets. Let fi denote the

fraction of the portfolio that is comprised of asset i. Assuming that no short selling is

allowed, fi≥0. Clearly, . 11

=∑=

N

iif

The portfolio return is given by the weighted sum of the returns on the individual

assets in the portfolio:

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∑=

=N

iiip rfr

1

. (2)

We have described asset returns as if they are known with certainty. However,

there is typically uncertainty surrounding the amount that will be received at the time that

an asset is sold. We can use a probability distribution to describe this unknown rate of

return. If return is normally distributed, then only two parameters – its expected return

and its standard deviation (or variance) - are needed to describe this distribution. The

expected return is the return around which the probability distribution is centered; it is the

expected value of the probability distribution of possible returns. The standard deviation

describes the dispersion of the distribution of possible returns.

2.1. Expected Portfolio Return

Suppose there are N assets with random returns r1,…, rN. The corresponding

expected returns are E(r1),…,E(rN). An investor wishes to create a portfolio of these N

assets, by investing a fraction fi of his wealth in asset i.

Using the properties of expectation, we may compute the expected portfolio

return using equation (2):

∑=

=N

iiip rEfrE

1)()( ,

i.e., the expected portfolio return is equal to the weighted sum of the expected returns of

its individual asset components.

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2.2. Portfolio Variance

The volatility of an asset’s return can be measured by its variance. Variance is

often adopted as a measure of an asset’s risk. If denotes the variance of asset i’s

return, then the variance of the portfolio’s return is given by:

2iσ

[ ]( )( )[ ][

∑ ∑∑∑∑

∑∑∑∑

= ≠=

=

=

=

==

==

+=

=

−−=

−−=

−=

N

i

N

ijj ijjii

N

i i

N

ji ijji

N

ji jjiiji

jN

j jjiN

i ii

N

i iiN

i iip

fff

ff

rrrrffE

rrfrrfE

rfrfE

11

21

2

1,

1,

11

211

2

))((

)()(

)(

σσ

σ

σ

] , (3)

where )( ii rEr = . Note that portfolio variance is a combination of the variance of the

returns of each individual asset in the portfolio plus their covariance.

3. Estimating an asset’s mean and variance

Of course, asset i’s rate of return and variance are not known and must be

estimated. These values can be estimated based upon historical data using standard

statistical methods. Alternatively, one can use a scenario-based approach. We describe

the two methods below.

3.1. Estimating statistics using historical data

To estimate statistics using historical data, one must collect several periods of

historical returns on the assets. The estimated average return on asset i is computed as

the sample average of returns on asset i, iX

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T

xX t

it

i

∑= ,

where xit is the historical return on asset i in period t and there are T periods of historical

data.

The variance of return on asset i is estimated by , the historical sample variance

of returns on investment i:

2is

1

)( 2

2

−

−=∑

T

Xxs t

iit

i .

For example, Table 1 contains the monthly closing stock prices and monthly returns for

Sun Microsystems and Continental Airlines for the months January 2004 through

February 2006. The first column of this table indicates the month, the second and third

columns contain the closing stock prices for SUN and Continental Airlines, respectively.

The fourth and fifth columns contain the monthly stock returns for SUN (XSUN,t ) and

Continental Airlines (XCAL,t), respectively; these columns were populated using Equation

(1). In this example, T=26.

Month

SUN Stock Price

CAL Stock Price

SUN Return

CAL Return

Jan-04 55.55 15.65 8.12% -5.04%Feb-04 61.66 14.97 11.00% -4.35%Mar-04 62.38 12.58 1.17% -15.97%Apr-04 63.00 10.66 0.99% -15.26%May-04 62.04 10.50 -1.52% -1.50%Jun-04 63.63 11.37 2.56% 8.29%Jul-04 68.17 8.75 7.13% -23.04%Aug-04 61.60 9.65 -9.64% 10.29%Sep-04 73.98 8.60 20.10% -10.88%Oct-04 74.76 9.24 1.05% 7.44%Nov-04 82.45 11.16 10.29% 20.78%Dec-04 81.71 13.75 -0.90% 23.21%

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Jan-05 88.04 10.50 7.75% -23.64%Feb-05 98.80 11.14 12.22% 6.10%Mar-05 105.00 12.00 6.28% 7.72%Apr-05 98.74 12.00 -5.96% 0.00%May-05 103.00 13.70 4.31% 14.17%Jun-05 114.14 13.32 10.82% -2.77%Jul-05 126.33 15.81 10.68% 18.69%Aug-05 75.35 13.28 -40.35% -16.00%Sep-05 78.20 9.70 3.78% -26.96%Oct-05 73.76 13.20 -5.68% 36.08%Nov-05 78.10 15.59 5.88% 18.11%Dec-05 79.34 21.27 1.59% 36.43%Jan-06 95.20 20.50 19.99% -3.62%Feb-06 74.35 23.79 -21.90% 16.05%

Table 1: Monthly closing stock prices and returns for Sun Microsystems and Continental Airlines

Table 2 shows the mean and standard deviation of returns for these two stocks,

based upon the 26 months of historical data. The average monthly return for SUN, SUNX ,

and the average monthly return for Continental, CALX , is computed as the arithmetic

average of the monthly returns in the fourth and fifth columns, respectively. An estimate

of the variance of monthly return on SUN’s (Continental’s) stock, is computed as the

variance of the returns in the fourth (fifth) column of Table 1. If variance is used as a

measure of risk, then Continental is a riskier investment since it has a higher volatility (its

variance is higher).

Expected monthly return Variance of return Standard Deviation SUN 2.30% 1.54 12.40% Continental Airlines 2.86% 3.04 17.43% Table 2: Expected historical monthly returns, variances, and standard deviations of returns

3.2. The scenario approach to estimating statistics

Sometimes, historical market conditions are not considered a good predictor of

future market conditions. In this case, historical data may not be a good source for

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estimating expected returns or risk. When historical estimates are determined to be poor

predictors of the future, one can consider a scenario approach.

The scenario approach proceeds as follows:

Define a set of S future economic scenarios and assign likelihood that

scenario s will occur. ∑ , since in the future the economy must be in exactly

one of these economic conditions. Next, define each asset’s behavior (its return) under

each of the defined economic scenarios. Asset i’s expected return is computed as:

)(sp

∈=

Sssp 1)(

∑= s ii srspr )()( , (4)

where is asset i’s return under scenario s. )(sri

Similarly, we compute the variance of return on asset i as:

∑ −=s iii rsrspv 2))()(( . (5)

For example, suppose we use the scenario approach to predict expected monthly

return on SUN stock. We have determined that the economy may be in one of three

states: weak, stable, or strong, with a likelihood of 0.3, 0.45, and 0.25, respectively. Table

3 indicates the forecasted monthly stock returns under each of these future economic

conditions:

Scenario (s) Likelihood ( ))(sp Return( ))(sri )(*)( srsp i

2))((*)( ii rsrsp −Weak economy 0.30 -15.00% -4.50% 1.03Stable economy 0.45 9.00% 4.05% 0.13Strong economy 0.25 16.00% 4.00% 0.39 3.55% 1.55

Table 3: Definition of future possible scenarios for states of the economy

The first and second columns in Table 3 indicate the economic scenario and

likelihood that the scenario will occur, respectively. The third column contains the

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expected return under each of the defined future scenarios. The fourth and fifth columns

contain intermediate computations needed to calculate the expected return and standard

deviation of returns on SUN stock, based upon Equations (3) and (4). Using these

equations we find that the expected monthly return on SUN stock is 3.55%, variance of

monthly return is 1.55, with corresponding standard deviation of 12.46%.

Comparing the estimates of mean and standard deviation of SUN’s monthly

return using the historical data approach versus the scenario-based approach we find that

while the estimates of volatility of return are close in value, the estimates of monthly

return differ significantly. (2.3% versus 3.55%.) In Section 7 we discuss the negative

impact that can result from portfolio allocation based upon incorrect parameter estimation.

Thus, care must be taken to determine the correct method and assumptions when

estimating these values.

4. Diversification

We now explore how a portfolio’s risk, as measured by the variance of the

portfolio return, can be reduced when stocks are added to the portfolio. This

phenomenon, whereby a portfolio’s risk is reduced when assets are added to the portfolio,

is known as diversification.

Portfolio return is the weighted average of the returns of the assets in the portfolio,

weighted by their appearance in the portfolio. However, portfolio variance (as derived in

equation (3)) is given by:

∑ ∑∑ = ≠=

=+

N

i

N

ijj ijjii

N

i i fff1

12

12 σσ .

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Namely, portfolio variance is comprised of two components. One component is

the variances of the individual assets in the portfolio and the other component is the

covariance between the returns on the different assets in the portfolio. Covariance of

return between asset i and j is the expected value of the product of the deviations of each

of the assets from their respective means. If the two assets deviate from their respective

means in identical fashions (i.e., both are above their means or below their means at the

same time) then the covariance is highly positive, and its contribution to portfolio

variance is highly positive. If the return on one asset deviates below its mean at the time

that the return on the other asset deviates above its mean, then the covariance is highly

negative which reduces the overall portfolio variance.

Let us explore the impact of the covariance term on overall portfolio variance. If

all of the assets are independent, then the covariance terms equal zero and only the

variance of the individual assets in the portfolio contribute to overall portfolio variance.

In this case, the variance formula is:

21

2i

N

i if σ∑ =.

If the investor invests an equal amount in each of the N independent assets, then the

portfolio variance is:

221

221

111)1( iiN

iiN

i NNNNσσσ == ∑∑ ==

,

where 2iσ is the average variance of the assets in the portfolio. As N gets larger, the

portfolio variance goes to zero. Thus, for a portfolio comprised only of independent

assets, when the number of assets in the portfolio is large enough the variance of the

portfolio return is zero.

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Now consider N assets that are not independent. Without loss of generality,

assume the assets appear in the portfolio with equal weight. Then, variance of portfolio

return is:

iji

N

i

N

ijj

iji

N

i

N

ijj iji

NN

N

NNNN

N

NN

σσ

σσ

σσ

11)1(

11

)1(1

2

11

2

11

22

−+=

−−

+=

+

∑ ∑

∑ ∑

= ≠=

= ≠=

(6)

The final equality in (6) reveals that as the number of assets in the portfolio

increases, the contribution of the first component (variance) becomes negligible – it is

diversified away – and the second term (covariance) approaches average covariance.

Thus, while increasing the number of assets in a portfolio will diversify away the

individual risk of the assets, the risk attributed to the covariance terms cannot be

diversified away.

As a numerical example, consider the SUN and Continental Airlines monthly

stock returns from Section 3.1. Average monthly return on Sun Microsystems

(Continental Airlines) stock from January 2004 through February 2006 was 2.3%

(2.86%). A portfolio comprised of equal investments in Sun Microsystems and

Continental Airlines yields an average monthly return of 2.58%. The variance of the

monthly returns on Sun Microsystems stock over the 26 months considered is 1.54. The

variance of the monthly returns on Continental Airlines stock over that same time period

is 3.04. However, for the portfolio consisting of equal investments in Sun Microsystems

and Continental Airlines, the variance of portfolio returns is 1.09. This represents a

significant reduction in risk from the risk associated with either of the stocks alone. The

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explanation lies in the value of the covariance between the monthly returns on these two

stocks; the value of the covariance is -0.12. Because the returns on the stocks have a

negative covariance, diversification reduces the portfolio risk.

5. Efficient frontier

The discussion in Section 4 illustrates the potential benefits of combining assets in

a portfolio. For a risk averse investor, diversification provides the opportunity to reduce

portfolio risk while maintaining a minimum level of return. An investor can consider

different combinations of assets, each of which has an associated risk and return.

This naturally leads one to question whether there is an optimal way to combine

assets. We address this question within the context of assuming that: (i) for a fixed return

investors prefer the lowest possible risk, that is, investors are risk averse, (ii) for a given

level of risk, investors always prefer the highest possible return. This property is referred

to as nonsatiation, and (iii) the first two moments of the distribution of an asset’s return

are sufficient to describe the asset’s character; there is no need to gather information

about higher moments such as skew.

Given these assumptions, the risk-return trade-off of portfolios of assets can be

graphically displayed by constructing a plot with risk (as measured by standard deviation)

on the horizontal axis and return on the vertical axis. We can plot every possible

portfolio on this risk-return space. The set of all portfolios plotted form a feasible region.

By the risk averse assumption, for a given level of return investors prefer portfolios that

lie as far to the left as possible, since these have the lowest risk. Similarly, by the

nonsatiation property for a given level of risk investors prefer portfolios that lie higher on

the graph since these yield a greater return. The upper left perimeter of the feasible

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region is called the efficient frontier. It represents the least risk combination for a given

level of return. The efficient frontier is concave.

Figure 1 shows a mean-standard deviation plot for 10,000 random portfolios

created from the thirty stocks in the Dow from 1986 through 1991. It represents the

feasible region of portfolios. Figure 2 is the efficient frontier for the stocks from the

Dow Industrials from 1985 through 1991. (Both figures were taken from the NEOS

Server for Optimization website.)

Figure 1: Return vs. standard deviation for 10,000 random portfolios from the Dow Industrials

Figure 2: Efficient frontier for Dow Industrials from 1985-1991

5.1. Risk-reward portfolio optimization

Markowitz (1952) developed a single period portfolio choice problem where the

objective is to minimize portfolio risk (variance) for a specified return on the portfolio.

For this model, it is assumed that all relevant information required by investors to make

portfolio decisions is captured in the mean, standard deviation, and correlation of assets.

This method for portfolio selection is often referred to as mean-variance optimization

since it trades off an investor’s desire for higher mean return against his aversion to

greater risk as measured by portfolio variance.

The Markowitz model for portfolio optimization is given by:

Minimize ∑ ji ijji ff,

σ

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Subject to: ∑==

N

i ii Rrf1

(7)

∑ =

=N

i if1

1

The objective is to minimize variance subject to two constraints: (i) portfolio

return must equal the targeted return R and (ii) total allocation must equal 1. Negative

values for fi correspond to short selling.

One method that can be employed to solve this constrained optimization problem

is to form an auxiliary function L called the Lagrangian by (i) rearranging each constraint

so that the right hand side equals to zero and (ii) introducing a single Lagrange multiplier

for each constraint in the problem as follows:

)1()(11, ∑∑∑ ==

−−−−=N

i iN

i iiji ijijji fRrfffL μλρσ . (8)

We now treat the Lagrangian (8) as an unconstrained minimization problem. A

necessary condition for a point to be optimal is that the partial derivates of the

Lagrangian with respect to each of the variables must equal zero. Thus, we take the

partial derivative of L with respect to each of the i asset weights fi, μ, and λ and set each

partial derivative equal to zero. (Notice that the partial derivative of L with respect to λ

yields the portfolio return constraint, and the partial derivative of L with respect to μ

yields the constraint on the asset weights.) The result is a system of i+2 constraints. We

use this set of constraints to solve for the i+2 unknowns: fi, μ, and λ.

The Lagrangian method can often only be successfully implemented for small

problems. For larger problems with many variables it may be virtually impossible to

solve the set of equations for the i+2 unknowns. Instead, one can find the solution to

Problem (7) using methods developed for optimizing quadratic programs. A quadratic

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program is a mathematical optimization model where the objective function is quadratic

and all constraints are linear equalities or inequalities. The constraints of the

optimization problem define the feasible region within which the optimal solution must

lie. Quadratic programs are, in general, difficult to solve. Quadratic programming

solution methods work in two phases. In the first phase a feasible solution is found. In

the second phase, the method searches along the edges and surfaces of the feasible region

for another solution that improves upon the current feasible solution. Unless the

objective function is convex, the method will often identify a local optimal solution.

The solution to the Markowitz problem yields a point that lies on the efficient

frontier. By varying the value of R in Problem (7), one can map out the entire efficient

frontier.

6. Utility analysis

The solution to the Markowitz problem provides one means for making

investment decisions in the mean-variance space. It requires the investor to define a

measure of risk and a measure of value and then utilizes an explicitly defined trade-off

between these two measures to determine the investor’s preference decisions. Utility

theory provides an alternative way to establish an investor’s preferences without

explicitly defining risk functions.

Utility describes an investor’s attitude toward risk by translating the investor’s

satisfaction associated with different cash payouts into a utility value. The application of

utility to uncertain financial situations was first introduced by von Neumann and

Morgenstern (1944). Utility functions can be used to explain how investors make choices

between different portfolios.

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Utility theory is often introduced by way of the concept of certainty equivalent.

Certainty equivalent is the amount of wealth that is equally preferred to an uncertain

alternative. Or, this certain amount has the same utility as the uncertain alternative. Risk

averse investors will prefer a lower certain cash payout to a higher risky cash payout.

That is, their certainty equivalent is lower than the expected value of uncertain

alternatives. Risk seeking individuals have a certainty equivalent that is higher than the

expected value of the uncertain alternatives. A utility function captures these attitudes

toward risk. It assigns different weights to different outcomes according to the risk

profile of the individual investor. The shape of a utility function is defined by the risk

profile of the investor.

Each individual investor may have a different utility function, since each investor

may have a different attitude toward risk. However, all utility functions U satisfy the

following properties:

• Nonsatiation: Utility functions must be consistent with more being preferred to

less. If x and y are two cash payouts and x>y, then U(x)>U(y). This property is

equivalent to stating that the first derivative of the utility function, with respect to

cash payout, is positive.

• Risk preference: Economic theory presumes that an investor will seek to

maximize the utility of his investment. However, all investors will not make

identical investment decisions because they will not all share the same attitudes

toward risk. Investors can be classified into three classes, according to their

willingness to accept risk: risk averse, risk neutral, or risk taking. Risk averse

investors invest in investments where the utility of the return exceeds the risk-free

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rate. If no such investment exists, the investor invests in risk-free investments.

The utility function of a risk averse investor is increasing and concave in the cash

payout ( ; the value assigned to each additional dollar received

decreases due to the risk averse nature of the individual. Risk-neutral investors

ignore risk when making investment decisions. They seek investments with a

maximum return, irrespective of the risk involved. The utility function of a risk

neutral individual is linear increasing in the cash payout; the same value is

assigned to each additional dollar. Risk-taking investors are more likely to invest

in investments with a higher risk involved. The utility function of a risk-taking

individual is convex ( .

)0)( <′′ xU

)0)( >′′ xU

• Changes in wealth: Utility functions also define how an investor’s investment

decisions will be affected by changes in his wealth. Specifically, if an investor has

a larger amount of capital will this change his willingness to accept risk? The

Arrow-Pratt absolute risk aversion coefficient, given by

)()()(

xUxUxA

′′′

−=

is a measure of an investor’s absolute risk aversion. )(xA′ measures changes in an

investor’s absolute risk aversion as a function of changes in his wealth. If

( ), the investor has increasing (decreasing) absolute risk

aversion and he will hold fewer (more) dollars in risky assets as his wealth

increases. If , the investor has constant absolute risk aversion and he

will hold the same dollar amount of risky assets as his wealth increases.

0)( >′ xA 0)( <′ xA

0)( =′ xA

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Unlike mean-variance optimization, where the cardinal value of the objective

function is meaningful, a utility function only ranks alternatives according to risk

preferences; its numerical value has no real meaning. Thus, utility functions are unique

up to a positive affine transformation. More specifically, if U(x) is a utility function then

V(x)=a+bU(x) (b>0) will yield the same rankings (and hence the same investment

decisions) as U(x).

6.1. Utility functions

Although each investor can define his own utility function, there are a number of

predefined utility functions that are commonly used in the finance and economics

literature. In this section we describe the exponential utility function. We also reconcile

between mean-variance optimization and utility optimization.

The exponential utility function is often adopted for financial analysis. The

exponential utility function is defined as:

Rx

exU−

−= 1)( , for all x (9)

where x is the wealth and R>0 represents the investor’s risk tolerance. Greater values of

R mean that the investor is less risk averse. (In fact, as R→∞ the investor becomes risk

neutral.) Rx

eR

xU−

=′ 1)( and Rx

eR

xU−

−=′′ 2)1()( . The second derivative of (9) is strictly

negative so the exponential utility function is concave; the exponential utility function

describes the behavior of a risk averse investor. The absolute risk aversion is )1(R

, which

is constant with wealth; the investor invests constant dollar amounts in risky investments

as his wealth increases.

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We now construct the relationship between utility and mean-variance for concave

(i.e., risk-averse) investors. The relationship can be constructed following King (1993).

Suppose U(X) is a strictly concave utility function. The Taylor expansion of U(X) is an

approximation of the function at a particular point, say M, using its derivatives. The

second order Taylor expansion of U(X) about point M is given by:

2))((21))(()()( MXMUMXMUMUXU −′′+−′+= . (10)

Now suppose that the point M is equal to expected wealth, i.e., E(X)=M. Then the

expected value of the second-order Taylor expansion expression (10) is equal to

2

2

)(21)(

)()(21)()]([

σMUMU

MXEMUMUXUE

′′+=

−′′+=

(The middle term drops out since E(X)=M.) The second derivative is negative for a

strictly concave utility function. This implies that maximizing the second order Taylor

expansion is equivalent to

22)(min σ=− MXE

for all X with E(X)=M.

However, this is equivalent to the mean-variance problem with a given mean

return M. It follows that mean-variance is a second-order approximation to utility

maximization. An investor who makes investment decisions by maximizing his expected

utility is equivalently performing a mean-variance optimization. Of course, due to the

two-sided nature of the variance, eventually this approximation will become negatively-

sloped – and hence not really valid as a utility function – as X increases in value.

However, the range over which the approximation is valid can be pretty wide. The upper

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bound of the range is the point where 0))(()( =−′′+′ MXXUXU , in other words this is

the maximum point of the approximating quadratic. This range can be quite wide enough

in practice. For example, for the logarithmic utility ),log()( XXU = the upper bound of

the range where the mean-variance approximation remains quadratic is In

other words, the mean-variance approximation for the logarithm is valid for a range that

includes twice the mean value of the return!

.2MX =

6.2. Utility in Practice

In practice, utility is not often used as an objective criterion for investment

decision making because utility curves are difficult to estimate. However, Holmer (1998)

reports that Fannie Mae uses expected utility to optimize its portfolio of assets and

liabilities. Fannie Mae faces a somewhat unique set of risks due to the specific nature of

its business. Fannie Mae buys mortgages on the secondary market, pools them, and then

sells them on the open market to investors as mortgaged backed securities. Fannie Mae

faces many risks such as: prepayment risk, risks due to potential gaps between interest

due and interest owed, and long term asset and liability risks due to interest rate

movements. Utility maximization allows Fannie Mae to explicitly consider degrees of

risk aversion against expected return to determine its risk adjusted optimal investment

portfolio.

7. Black - Litterman Asset Allocation Model

Markowitz mean-variance portfolio optimization requires mean and covariance as

input and outputs optimal portfolio weights. The method has been criticized because:

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(i) The optimal portfolio weights are highly dependent upon the input values.

However, it is difficult to accurately estimate these input values. Chopra and

Ziemba (1993), Kallberg and Ziemba (1981, 1984), and Michaud (1989) use

simulation to demonstrate the significant cash-equivalent losses due to incorrect

estimates of the mean. Bengtsson (2004) showed that incorrect estimates of

variance and covariance also have a significant negative impact on cash returns.

(ii) Markowitz mean-variance optimization requires the investor to specify the

universe of return values. It is unreasonable to expect an investor to know the

universe of returns. On the other hand, mean-variance optimization is sensitive to

the input values so incorrect estimation can significantly skew the results.

(iii) Black and Litterman (1992) and He and Litterman (1999) have studied the

optimal Markowitz model portfolio weights assuming different methods for

estimating the assets means’ and found that the resulting portfolio weights were

unnatural. Unconstrained mean-variance optimization typically yields an optimal

portfolio that takes many large long and short positions. Constrained mean-

variance optimization often results in an extreme portfolio that is highly

concentrated in a small number of assets. Neither of these portfolio profiles is

typically considered acceptable to investors.

(iv) Due to the intricate interaction between mean and variance, the optimal weights

determined by Markowitz’s mean-variance estimation are often non-intuitive. A

small change in an estimated mean of a single asset can drastically change the

weights of many assets in the optimal portfolio.

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Black and Litterman observed the potential benefit of using mathematical

optimization for portfolio decision-making, yet understood investment manager’s

hesitations in implementing Markowitz’s mean-variance optimization model. Black and

Litterman (1992) developed a Bayesian method for combining individual investor

subjective views on asset performance with market equilibrium returns to create a mixed

estimate of expected returns. The Bayes approach works by combining a prior belief

with additional information to create an updated “posterior” distribution of expected asset

returns. In the Black-Litterman framework the equilibrium returns are the prior and

investor subjective views are the additional information. Together, these form a posterior

distribution on expected asset returns. These expected returns can then be used to make

asset allocation decisions. If the investor has no subjective views on asset performance,

then the optimal allocation decision is determined solely according to the market

equilibrium returns. Only if the investor expresses opinions on specific assets will the

weights for those assets shift away from the market equilibrium weights in the direction

of the investor’s beliefs.

The Black-Litterman model is based on taking a market equilibrium perspective

on asset returns. Asset “prior” returns are derived from the market capitalization weights

of the optimal holdings of a mean-variance investor, given historical variance. Then, if

the investor has specific views on the performance of any assets, the model combines the

equilibrium returns with these views, taking into consideration the level of confidence

that the investor associates with each of the views. The model then yields a set of

updated expected asset returns as well as updated optimal portfolio weights, updated

according to the views expressed by the investor.

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The key inputs to the Black-Litterman model are market equilibrium returns and

the investor views. We now consider these inputs in more detail.

7.1. Market equilibrium returns

Black and Litterman use the market equilibrium expected returns, or CAPM

returns, as a neutral starting point in their model. (See, e.g., Sharpe (1964).) The basic

assumptions are that (i) security markets are frictionless, (ii) investors have full

information relevant to security prices, and (iii) all investors process the information as if

they were mean-variance investors. The starting point for the development of the CAPM

is to form the efficient frontier for the market portfolios and to draw the Capital Market

Line (CML). The CML begins at the risk free rate on the vertical axis (which has risk 0)

and is exactly tangent to the efficient frontier. The point where the CML touches the

efficient frontier is the pair ),( mm rσ , which is defined to be the “market” standard

deviation and “market” expected return. By changing the relative proportions of riskless

asset and market portfolio, an investor can obtain any combination of risk and return that

lies on the CML. Because the market point is tangent to the line, there are no other

combinations of risky and riskless assets that can provide better expected returns for a

given level of risk. Now consider a particular investor portfolio i with expected return

E(ri) and standard deviation iσ . For an investor to choose to hold this portfolio it must

have returns comparable to the returns that lie on the Capital Market Line. Thus, the

following must hold:

)()( fmm

ifi rrrrE −+=

σσ

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where rf is the risk free rate. This equation is called the Capital Asset Pricing Model, or

CAPM.

The interpretation of the CAPM is that investor’s portfolios have an expected

return that includes a reward for taking on risk. This reward, by the CAPM hypothesis,

must be equal to the return that would be obtained from holding a portfolio on the Capital

Market Line that has an equivalent risk. Any remaining risk in the portfolio can be

diversified away (by, for example, holding the market portfolio) so the investor does not

gain any reward for the non-systematic, or diversifiable, risk of the portfolio. The CAPM

argument applied to individual securities implies that the holders of individual securities

will be compensated only for that part of the risk that is correlated with the market, or the

so-called systematic risk. For an individual security j the CAPM relationship is

)()( fmjfj rrrrE −+= β

where mjm

jj ρ

σσ

β = and mjρ is the correlation between asset j returns and the market

returns.

The Black-Litterman approach uses the CAPM in reverse, by assuming that in

equilibrium the market portfolio is held by mean-variance investors and by using

optimization to back out the expected returns that such investors would require given

their observed holdings of risky assets. Let N denote the number of assets and let the

excess equilibrium market returns (above the risk free rate) be defined by:

w∑=Π λ , (11)

where

П = Nx1 vector of implied excess returns

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Σ = NxN covariance matrix of returns

w = Nx1 vector of market capitalization weights of the assets

λ = risk aversion coefficient that characterizes the expected risk-reward tradeoff.

λ is the price of risk as it measures how risk and reward can be traded off when

making portfolio choices. It measures the rate at which an investor will forego expected

return for less variance. λ is calculated as λ=(rm- rf)/σ2m, where σ2

m is the variance of the

market return. The elements of the covariance matrix are computed using historical

correlations and standard deviations. Market CAP weights are determined by measuring

the dollar value of the global holdings of all equity investors in the large public stock

exchanges. The CAP weight of a single equity name is the dollar-weighted market-

closing value of its equity share times the outstanding shares issued. Later we will show

that Equation (11) is used to determine the optimal portfolio weights in the Black-

Litterman model.

7.2. Investor views

The second key input to the Black-Litterman model is individual investor views.

Assume that an investor has K views, denoted by a Kx1 vector Q. Uncertainty

regarding these views is denoted by an error term ε, where ε is normally distributed with

mean zero and KxK covariance matrix Ω. Thus, a view has the form:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=+

KKQ

Q

Q

ε

ε

εMM11

.

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ε=0 means that the investor is 100% confident about his view; in the more likely case

that the investor is uncertain about his view, ε takes on some positive or negative value.

ω denotes the variance of each error term. We assume that the error terms are

independent of each other. (This assumption can be relaxed.) Thus, the covariance

matrix Ω is a diagonal matrix where the elements on the diagonal are ω, the variances of

each error term. A higher variance indicates greater investor uncertainty with the

associated view. The error terms ε do not enter directly into the Black-Litterman

formula; only their variances enter via the covariance matrix Ω.

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Ω

Kω

ω

000

0001

K

M

MO

K

.

The Black-Litterman model allows investors to express views such as:

View 1: Asset A will have an excess return of 5.5% with 40% confidence.

View 2: Asset B will outperform asset C by 3% with 15% confidence.

View 3: Asset D will outperform assets E and F by 1% with 20% confidence.

The first view is called an absolute view, while the second and third views are called

relative views. Notice that the investor assigns a level of confidence to each view.

Each view can be seen as a portfolio of long and short positions. If the view is an

absolute view then the portfolio position will be long. If the view is a relative view, then

the portfolio will take a long position in the asset that is expected to “overperform” and a

short position in the asset that is expected to “underperform.” In general, the impact on

the optimal portfolio weights is determined by comparing the equilibrium difference in

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performance of these assets to the performance expressed by the investor view. If the

performance expressed in the view is better than the equilibrium performance, the model

will tilt the portfolio toward the outperforming asset. More specifically, consider View 2

which states that asset B will outperform asset C by 3%. If the equilibrium returns

indicate that asset B will outperform asset C by more than 3% then the view represents a

weakening view in performance of asset B and the model will tilt the portfolio away from

asset B.

One of the most challenging questions in applying the Black-Litterman model is

how to populate the covariance matrix Ω and how to translate the user specified

expressions of confidence into uncertainty in the views. We will discuss this further

below.

7.3. An example of investor view

We now illustrate how one builds the inputs for the Black-Litterman model, given

the three views expressed. Suppose that there are N=7 assets: Assets A-H. The Q matrix

is given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

135.5

Q .

Note that the investor only has views on six of the seven assets. We use the

matrix P to match the views to the individual assets. Each view results in a 1xN vector so

that P is a KxN matrix. In our case, where there are seven assets and three views, P is a

3x7 matrix. Each column corresponds to one of the assets; column 1 corresponds to

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Asset A, column 2 corresponds to Asset B, etc. In the case of absolute views, the sum of

the elements in the row equals 1. In our case, View 1 yields the vector:

[ ]00000011 =P .

In the case of relative views, the sum of the elements equals zero. Elements

corresponding to relatively outperforming assets have positive values; elements

corresponding to relatively underperforming assets take negative values. We determine

the values of the individual elements by dividing 1 by the number of outperforming and

underperforming assets, respectively. For view 2, we have one outperforming asset and

one underperforming asset. Thus, View 2 yields the vector:

[ ]00001102 −=P .

View 3 has one outperforming asset (Asset D) and two relatively

underperforming assets (Assets E and F). Thus, Asset D is assigned a value of +1 and

Assets E and F are assigned values of -0.5 each. View 3 yields the vector:

[ ]05.05.010003 −−=P .

Matrix P is given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

05.05.0100000001100000001

.

The variance of the kth view portfolio can be calculated according to the formula

, where is the kkk pp 'Σ kp th row of the P matrix and Σ is the covariance matrix of the

excess equilibrium market returns. (Recall, these form the neutral starting point of the

Black-Litterman model). The variance of each view portfolio is an important source of

information regarding the confidence that should be placed in the investor’s view k.

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7.4. Combining equilibrium returns with investor views

We now state the Black-Litterman equation for combining equilibrium returns

with investor views to determine a vector of expected asset returns that will be used to

determine optimal portfolio weights. The vector of combined asset returns is given by:

])[(])[(][ 11111 QPPPRE −−−−− Ω′+ΠΣΩ′+Σ= ττ (12)

where:

E[R] = Nx1 vector of combined returns

τ = scalar, indicating uncertainty of the CAPM prior

Σ = NxN covariance matrix of equilibrium excess returns

P = KxN matrix of investor views

Ω = KxK diagonal covariance matrix of view error terms (uncorrelated view

uncertainty)

Π= Nx1 vector of equilibrium excess returns

Q = Kx1 vector of investor views

Examining this formula, we have yet to describe how the value of τ should be set

and how the matrix Ω should be populated. Recall that if an investor has no views, the

Black-Litterman model suggests that the investor does not deviate from the market

equilibrium portfolio. Only weights on assets for which the investor has views should

change from their market equilibrium weights. The amount of change depends uponτ ,

the investor’s confidence in the CAPM prior, and ω, the uncertainty in the views

expressed.

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The literature does not have a single view regarding how the value ofτ should be

set. Black and Litterman (1992) suggest a value close to zero. He and Litterman (1999)

set τ equal to 0.025 and populate the covariance matrix Ω so that

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

′Σ

′Σ

=Ω

τ

τ

)(000

000)( 11

KK pp

pp

K

M

MO

K

.

We note that the implied assumption is that the variance of the view portfolio is the

information that determines an investor’s confidence in his view. There may be other

information that contributes to the level of an investor’s confidence but it is not

accounted for in this method for populating Ω .

Formula (12) uses Bayes approach to yields posterior estimates of asset returns

that reflect a combination of the market equilibrium returns and the investor views.

These updated returns are now used to compute updated optimal portfolio weights.

In the case that the investor is unconstrained, we use formula (11). Using formula

(11) , the optimal portfolio weights, are given by: *w

μλ 1* )( −Σ=w , (13)

whereμ is the vector of combined returns. Equation (13) is the solution to the

unconstrained maximization problem 2/max wwww Σ′−′ λμ .

In the presence of constraints (e.g., risk, short selling, etc.) Black and Litterman

suggest that the vector of combined returns be input into a mean-variance.

We note two additional comments on the updated weights : *w

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(i) Not all view portfolios necessarily have equal impact on the optimal portfolio

weights derived using the Black-Litterman model. A view with a higher level of

uncertainty is given less weight. Similarly, a view portfolio that has a covariance with

the market equilibrium portfolio is given less weight. This is because such a view

represents less new information and hence should have a smaller impact in moving the

optimal portfolio weights away from the market equilibrium weights. Finally, following

the same reasoning, a view portfolio that has a covariance with another view portfolio has

less weight.

(ii) A single view causes all returns to change, because all returns are linked via

the covariance matrixΣ . However, only the weights for assets for which views were

expressed change from their original market capitalization weights. Thus, the Black-

Litterman model yields a portfolio that is intuitively understandable to the investor. The

optimal portfolio represents a combination of the market portfolio and a weighted sum of

the view portfolios expressed by the investor.

7.5. Application of Black-Litterman Model

The Black-Litterman model was developed at Goldman Sachs in the early 1990s

and is used by the Quantitative Strategies group at Goldman Sachs Asset Management.

This group develops quantitative models to manage portfolios. The group creates views

and then uses the Black-Litterman approach to transform these views into expected asset

returns. These expected returns are used to make optimal asset allocation decisions for

all of the different portfolios managed by the group. Different objectives or requirements

(such as liquidity requirements, risk aversion, etc.) are incorporated via constraints on the

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portfolio. The Black-Litterman model has gained widespread use in other financial

institutions.

8. Risk Management

Risk exists when more than one outcome is possible from the investment.

Sources of risk may include business risk, market risk, liquidity risk, and the like.

Variance or standard deviation of return is often used as a measure of the risk associated

with an asset’s return. If variance is small, there is little chance that the asset return will

differ from what is expected; if variance is large then the asset returns will be highly

variable.

Financial institutions manage their risk on a regular basis both to meet regulatory

requirements as well as for internal performance measurement purposes. However, while

variance is a traditional measure of risk in economics and finance, in practice it is

typically not the risk measure of choice. Variance assumes symmetric deviations above

and below expected return. In practice, one does not observe deviations below expected

return as often as deviations above expected return due to positions in options and

options-like instruments in portfolios. Moreover, variance assigns equal penalty to

deviations above and below the mean return. However, investors typically are not averse

to receiving higher than anticipated returns. Investors are more interested in shortfall risk

measures. These are risk measures that measure either the distance of return below a

target or measure the likelihood that return will fall below a threshold.

One measure of shortfall risk is downside risk. Downside risk measures the

expected amount by which the return falls short of a target. Specifically, if z is the

realized return and X is the target then downside risk is given by [ ])0,max( zXE − .

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Semivariance is another measure of shortfall risk. Semivariance measures the variance of

the returns that fall below a target value. Semivariance is given by [ ]2)0,max( zXE − .

8.1. Value at Risk Value at Risk (VaR) is a risk measure that is used for regulatory reporting. Rather

than measuring risk as deviation from a target return, VaR measures risk as required

capital. VaR is a risk measurement on the loss distribution of a portfolio. Let be

the random loss on a portfolio with allocation vector f and random return vector

)~,( rfL

r~ . Let

F be its distribution function so that urfLufF ≤= )~,(Pr),( . Value at risk is the α-

quantile of the loss distribution and is defined by:

αςςαςςςα −≥≥=≥= 1)~,(:min),(:min),( rfLPfFfVaR , (14)

where ζ is the value at risk and r~ are the random asset returns. Thus, value-at-risk is the

smallest amount u such that with probability α the loss will not exceed u.

The first step in computing VaR is to determine the underlying market factors that

contribute to the risk (uncertainty) in the portfolio. The next step is to simulate these

sources of uncertainty and the resulting portfolio loss. Monte Carlo simulation is used

largely because many of the portfolios under management by the more sophisticated

banks include a preponderance of instruments that have optional features. As we shall

see in Section 10, the price changes of these instruments can best be approximated by

simulation. VaR can then be calculated by determining the distribution of the portfolio

losses. The key question is what assumptions to make about the distributions of these

uncertain market factors. Similar to the two methods that we discuss for estimating asset

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returns and variances, one can use historical data or a scenario approach to build the

distributions.

Using historical data, one assumes that past market behavior is a good indicator of

future market behavior. Take T periods of historical data. For each period, simulate the

change in the portfolio value using the actual historical data. Use these T data points of

portfolio profit/loss to compute the loss distribution and hence VaR. The benefit of this

method is that there is no need for artificial assumptions about the distribution of the

uncertainty of the underlying factors that impact the value of the portfolio. On the other

hand, this method assumes that future behavior will be identical to historical behavior.

An alternative approach is to specify a probability distribution for each of the

sources of market uncertainty and to then randomly generate events from those

distributions. The events translate into behavior of the uncertain factors, which result in a

change in the portfolio value. One would then simulate the portfolio profit/loss assuming

that these randomly generated events occur and construct the loss distribution. This

distribution is used to compute VaR.

8.2. Coherent Risk Measures

VaR is a popular risk measure. However, VaR does not satisfy one of the basic

requirements of a good risk measure: VaR is not subadditive for all distributions (i.e., it is

not always the case that VaR(A+B)<VaR(A)+VaR(B)), a property one would hope to hold

true if risk is reduced by adding assets to a portfolio. This means that the VaR of a

diversified portfolio may exceed the sum of the VaR of its component assets.

Artzner, Delbaen, Eber, and Heath (1999) specify a set of axioms satisfied by all

coherent risk measures. These are:

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• Subadditivity: ρ(A+B) ≤ρ(A)+ρ(B)

• Positive Homogeneity: ρ(λA)= λρ(A) for λ≥0.

• Translation invariance: ρ(A+c) =ρ(A)-c for all c.

• Monotonicity: A≤B then ρ(B) ≤ ρ(A)

Subadditivity implies that the risk of a combined position of assets does not

exceed the combined risk of the individual assets. This allows for risk reduction via

diversification, as we discuss in Section 4.

Conditional Value-at-risk (CVaR), also known as Expected Tail Loss, is a

coherent risk measure. CVaR measures the expected losses conditioned on the fact that

the losses exceed VaR. Following the definition of VaR in equation (14), if F is

continuous then CVaR is defined as:

),() ~,(|)~,(),( αα fVaRrfLrfLEfCVaR ≥= . (15)

An investment strategy that minimizes CVaR will minimize VaR as well.

An investor wishing to maximize portfolio return subject to a constraint on

maximum CVaR would solve the following mathematical program:

10

1

),...,(:

)(max

1

1

1

≤≤

=

≤

∑

∑

=

=

i

N

i i

N

N

iii

f

f

CffCVaRtosubject

rfE

α (16)

where fi is the fraction of wealth allocated to asset i, ri is the return on asset i, and C is the

maximum acceptable risk. Formulation (16) is a nonlinear formulation due to the

constraint on CVaR, and is a hard problem to solve.

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Suppose, instead of assuming that the loss distribution F is continuous, we

discretize the asset returns by considering a finite set of s=1,…,S scenarios of the

portfolio performance. Let p(s) denote the likelihood that scenario s occurs;

. 1)(;1)(0 =≤≤ ∑∈Ss

spsp )(sϕ denotes the vector of asset return under scenario s and

))(,( sf ϕρ denotes the portfolio return under scenario s assuming asset allocation vector f.

Then, using this discrete set of asset returns, the expected portfolio return is given by

∑ ∈Sssfsp ))(,()( ϕρ . (17)

The investor will wish to maximize (17). Definition (15) applies to continuous loss

distributions. Rockafellar and Uryasev (2002) have proposed an alternative definition of

CVaR for any general loss distribution:

∑∑≤∈≤∈ −

+−

−=))(,(|))(,(|

))(,()(1

1))(1

11(),(ςϕρςϕρ

ϕρα

ςα

αsfSssfSs

sfspspfCVAR (18)

Using definition (18) we can solve problem (16) using a stochastic programming

approach. First, we define an auxiliary variable zs for each scenario s, which denotes the

shortfall in portfolio return from the value at risk ζ:

)))(,(,0max()( sfsz ϕρς −= (19)

Following Rockafellar and Uryasev (2002), CVaR can be expressed using the shortfall

variables (19) as:

∑ =−−=

S

sszspfCVaR

1)()(

11),(α

ςα .

The linear program is given by:

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SsszSssfsz

Cszsp

f

f

tosubject

sfsp

S

s

i

N

i i

Ss

,...,10)(,...,1))(,()(

)()(1

110

1

:

))(,()(max

1

1

=≥=−≥

≥−

−

≤≤

=

∑

∑

∑

=

=

∈

ϕρςα

ς

ϕρ

(20)

where the last two inequalities follow from the definition of the shortfall variables (19).

Formulation (20) is a linear stochastic programming formulation of the CVaR

problem. To solve this problem requires an estimate of the distribution of the asset

returns, which will be used to build the scenarios. If historical data is used to develop the

scenarios then it is recommended that as time passes and more information is available,

the investor reoptimize problem (20) (rebalance his portfolio) using these additional (in-

sample) scenarios. We direct the reader to Rockafellar and Uryasev (2000,2002) for

additional information on this subject.

8.3. Risk Management in Practice

VaR and CVaR are popular risk measures. VaR is used in regulatory reporting

and to determine the minimum capital requirements to hedge against market, operational,

and credit risk. Financial institutions may be required to report portfolio risks such as the

30-day 95% VaR or the 5% quantile of 30-day portfolio returns and to hold reserve

accounts in proportion to these calculated amounts. VaR is used in these contexts for

historical reasons. But even though as we saw above it is not a coherent risk measure,

there is possibly some justification in continuing to use it. VaR is a frequency measure,

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so regulators can easily track whether the bank is reporting VaR accurately; CVaR is an

integral over the tail probabilities and is likely not as easy for regulators to track.

In addition to meeting regulatory requirements, financial institutions may use VaR or

CVaR to measure the performance of its business units that control financial portfolios by

comparing the profit generated by the portfolio actions versus the risk of the portfolio

itself.

For purposes of generating risk management statistics, banks will simulate from

distributions that reflect views of the market and the economy. Banks will also

incorporate some probability of extreme events such as the wild swings in correlations

and liquidity that occur in market crashes.

9. Options

In this section we discuss options. An option is a derivative security, which

means that its value is derived from the value of an underlying variable. The underlying

variable may or may not be a traded security. Stocks or bonds are examples of traded

securities; interest rates or the weather conditions are examples of variables upon which

the value of an option may be contingent but that are not traded securities. Derivative

securities are sometimes referred to as contingent claims, since the derivative represents a

claim whose payoff is contingent on the history of the underlying security.

The two least complex types of option contracts for individual stocks are calls and

puts. Call options give the holder the right to buy a specified number of shares (typically,

100 shares) of the stock at the specified price (known as the exercise or strike price) by

the expiration date (known as the exercise date or maturity). A put option gives the

holder the right to sell a specified number of shares of the stock at the strike price by

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maturity. American options allow the holder to exercise the option at any time until

maturity; European options can only be exercised at maturity. The holder of an option

contract may choose whether or not he wishes to exercise his option contract. However, if

the holder chooses to exercise, the seller is obligated to deliver (for call options) or

purchase (for put options) the underlying securities.

When two investors enter into an options contract, the buyer pays the seller the

option price and takes a long position; the seller takes a short position. The buyer has

large potential upside from the option, but his downside loss is limited by the price that

he paid for the option. The seller’s profit or loss is the reverse of that of the seller’s. The

seller receives cash upfront (the price of the option) but has a potential future liability in

the case that the buyer exercises the option.

9.1. Call option payoffs

We first consider European options. We will define European option payoffs at

their expiration date.

Let C represent the option cost, ST denote the stock price at maturity, and K

denote the strike price. An investor will only exercise the option if the stock price

exceeds the strike price, i.e., ST >K. If ST >K, the investor will exercise his option to buy

the stock at price K and gain (ST –K) for each share of stock that he purchases. If ST <K,

then the investor will not exercise the option to purchase the stock at price K. Thus, the

option payoff for each share of stock is

max(ST -K,0). (21)

The payoff for the option writer (who has a short position in a call option) is the

opposite of the payoff for the long position. If, at expiration, the stock price is below the

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strike price the holder will not exercise the option. However, if the stock price is above

the strike price the owner will exercise his option. The writer must sell the stock to the

owner at the strike price. For each share of stock that he sells, the writer must purchase

the stock in the open market at a price per share of ST and then sell it to the owner at a

price of K. Thus, the writer loses (ST –K) for each share that he is obligated to sell to the

owner. The writer thus has an unlimited potential loss depending upon the final stock

price of the underlying asset; the writer’s payoff is

-max(ST -K,0)= min(K-ST,0).

Thus, an investor with a short position in a European call option has potentially unlimited

loss depending upon the final stock price of the underlying stock. This risk must be

compensated by the price of the option .C

The graph on the left of Figure 3 depicts the profit at maturity for the owner of a

call option. The point of inflection occurs when the ending stock price equals the strike

price. The negative payoff is the price the investor paid for the option. As mentioned,

the option holder’s downside loss is limited by the price paid for the option. The payoff

for the call option writer is depicted in the graph on the right of Figure 3 and is the

reverse of the payoff to the call option buyer.

Figure 3 about here

Figure 3: Profit from European Call Options

When the price of the underlying stock is above the strike price, we say that the

option is “in the money.” If the stock price is below the strike price we say that the

option is “out of the money.” If the stock price is exactly equal to the strike price we say

that the call option is “at the money.”

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American options can be exercised at any time prior to maturity. The decision of

whether or not to exercise hinges upon a comparison of the value of exercising

immediately (the intrinsic value of the option) against the expected future value of the

option if the investor continues to hold the options. We will discuss this in further detail

in Section 12 when we discuss pricing American options.

9.2. Put option payoffs

An investor with a long position in a put option profits if the price of the

underlying stock drops below the option’s strike price. Similar to the definitions for a

call option, we say that a put option is “in the money” if the stock price at maturity is

lower than the strike price. The put option is “out of the money” if the stock price

exceeds the strike price. The option is “at the money” if the strike price equals the stock

price at maturity.

Let P denote the cost of the put option, K is its strike price, and ST the stock price

at expiration. The holder of a put option will only sell if the stock price at expiration is

lower than the strike price. In this case, the owner can sell the shares to the writer at the

strike price and will gain (K- ST) per share. Thus, payoff on a put option is max (K- ST,0);

we note that positive payoff on a put is limited at K. If the stock price is higher than the

strike price at maturity then the holder will not exercise the put option since he can sell

his shares on the open market at a higher price. In this case, the option will expire

worthless.

A put writer has opposite payoffs. If the stock price exceeds the strike price, the

put holder will never exercise his option, however, if the stock price declines, the writer

will lose money. The price P paid by the owner must compensate the writer for this risk.

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Figure 4 depicts the option payoff for a put holder and writer.

Figure 4 about here

Figure 4: Profit from European put options

10. Valuing options

The exact value of a stock option is easy to define at maturity. Valuing options

prior to expiration is more difficult and depends upon the distribution of the underlying

stock price, amongst other factors. Black and Scholes (1973) derived a differential

equation that can be used to price options on non-dividend paying stocks. We discuss the

Black-Scholes formula in Section 10.1. However, in general exact formulas are not

available for valuing options. In most cases, we rely on numerical methods and

approximations for options valuation. In Sections 1010.3 and 1112 we discuss two

numerical methods for pricing options: Monte Carlo simulation and dynamic

programming. Dynamic programming is useful for pricing American options, where the

holder has the ability to exercise prior to the expiration date. Monte Carlo simulation is

useful for pricing a European option, where the option holder cannot exercise the option

prior to the maturity date. In the following sections we describe Monte Carlo simulation

and dynamic programming and show how they can be used to price options. We first

will lay out some background and basic assumptions that are required.

10.1. Risk Neutral Valuation in efficient and complete

markets

We base our discussion of options pricing on the assumption that markets are

efficient and complete. Efficient markets are arbitrage-free. Arbitrage provides an

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investor with a riskless investment opportunity with unlimited return, without having to

put up any money. We assume that if any such opportunities exist there would be infinite

demand for such assets. This would immediately raise the price of the investments and

the arbitrage opportunity would disappear.

Black and Scholes derived a differential equation that describes the value of a

trader’s portfolio who has a short position in the option and who is trading in the

underlying asset and a cash-like instrument. Efficiency of markets is one of the key

assumptions required in their derivation. In addition, they assume that instantaneous and

continuous trading opportunities exist, no dividends, transaction costs, or taxes are paid,

and that short selling is permitted. Finally, they assume that the price of the underlying

stock follows a specific stochastic process called Brownian motion. (See Section 10.2 for

discussion of Brownian motion.) In this Black-Scholes framework it turns out that there

is a trading strategy (called “delta-hedging”) that makes the portfolio’s return completely

riskless. In an efficient market, a riskless portfolio will return the risk-free rate. This

arbitrage reasoning combined with the delta-hedging strategy leads to a partial

differential equation that resembles the heat equation of classical physics. Its solution

provides the option’s value at any point in time. In the case of European-style options

(those that have a fixed exercise date) the solution can be achieved in closed form – this

is the famous Black-Scholes formula. The Black-Scholes formula for the values of a

European call C or put P are:

)()(

)()(

12

21

dSdKeP

dKedSCrT

rT

−Φ−−Φ=

Φ−Φ=−

−

where:

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Tdd

TrTKSd

σ

σσ

−=

++=

12

21 /]2/)()/[log(

Here, r is the risk-free rate (the rate of return of a riskless security such as a US Treasury

security over timeT ), log denotes the natural logarithm, and ()Φ is the cumulative

distribution function for the standard normal distribution . )1,0(N

Exotic option contracts, especially those with exercise rules that give the owner the

discretion of when to exercise, or options with underlying assets that are more

complicated than equity stocks with no dividends, or options that depend on multiple

assets, turn out to be very difficult to solve using the Black-Scholes partial differential

equation.

Harrison and Pliska (1981) developed a more general perspective on options

pricing that leads to a useful approach for these more general categories of options. It

turns out that the existence of the riskless trading strategy in the Black-Scholes

framework can be viewed as mathematically equivalent to the existence of a dual object

called a “risk-neutral measure”, and also that the options price is the integral of the option

payouts with this risk-neutral measure. When the risk-neutral measure is unique, the

market is called “complete”. This assumption means that there is a single risk-neutral

measure that can be used to price all the options.

This perspective leads to the following methodology for options pricing. Observe

the prices of the traded options. Usually these are of fairly simple type (European or

American calls and puts), for which closed-form expressions like the Black-Scholes

formula can be used. Then invert the formula to find the parameters of the risk-neutral

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distribution. This distribution can then be used to simulate values of any option – under

the assumption that the market is efficient (arbitrage-free) and complete.

10.2. Brownian Motion

A key component of valuing stock options is a model of the price process of the

underlying stock. In this section, we describe the Brownian motion model for the stock

prices.

The efficient market hypothesis, which states that stock prices reflect all history

and that any new information is immediately reflected in the stock prices, ensures that

stock prices follow a Markov process so the next stock price depends only upon the

current stock price and does not depend upon the historical stock price process. A

Markov process is a stochastic process with the property that only the current value of the

random variable is relevant for the purposes of determining the next value of the variable.

A Wiener process is a type of Markov process. A Wiener process Z(t) has normal and

independent increments with variance proportional to the square root of time, i.e., Z(t)-

Z(s) has a normal distribution with mean zero and variance st − . It turns out that Z(t),

t>0 will be a continuous function of t. If Δt represents an increment in time and ΔZ

represents the change in Z over that increment in time then the relationship between ΔZ

and Δt can be expressed by:

ttZ Δ=Δ ε)( (22)

where ε is drawn from a standard normal distribution. A Wiener process is the limit of

the above stochastic process as the time increments get infinitesimally small, i.e., as

Δt→0. Equation (22) is expressed as

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dttdZ ε=)( . (23)

If x(t) is a random variable and Z is a Wiener process, then a generalized Wiener process

is defined as

bdZadttdx +=)(

where a and b are constants. An Ito process is a further generalization of a generalized

Wiener process. In an Ito process, a and b are not constants rather, they can be functions

of x and t. An Ito process is defined as

dZtxbdttxatdx ),(),()( +=

Investors are typically interested in the rate of return on a stock, rather than the absolute

change in stock price. Let S be the stock price and consider the change in stock price dS

over a small period of time dt. The rate of return on a stock, dS/S, is often modeled as

being comprised of a deterministic and stochastic component. The deterministic

component, μdt, represents the contribution of the average growth rate of the stock. The

stochastic component captures random changes in stock price due to unanticipated news.

This component is often modeled by σdZ, where Z is a Brownian motion. Combining the

deterministic growth rate (also known as drift) with the stochastic contribution to rate of

change in stock price yields the equation:

dZdtSdS σμ +=/ , (24)

an Ito process. μ and σ can be estimated using the methods described in Section 3.

The risk-neutral measure of Harrison and Pliska as applied to the Black-

Scholes framework is induced by the risk-neutral process X that satisfies the modified

Brownian motion

Q

dZdtrXdX σσ +−= )2/(/ 2 (25)

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It is important to note that this process is not the same as the as the original process

followed by the stock – the drift has been adjusted. This adjustment is what is required to

generate the probability measure that makes the delta-hedging portfolio process into a

martingale. According to the theory discussed above in Section 10.1, we price options in

the Black-Scholes framework by integrating their cash flows under the risk-neutral

measure generated by equation (25). In the following section we discuss how efficient

markets and risk neutral valuation are used to compute an option’s value using Monte

Carlo simulation.

10.3. Simulating risk-neutral paths for options pricing

In this section we discuss how simulation can be used to price options on a stock

by simulating the stock price under the risk-neutral measure over T periods, each of

length Δt=1/52. At each time interval, we simulate the current stock price and then step

the process forward so that there are a total of T steps in the simulation. To simulate the

path of the stock price over the T week period, we consider the discrete time version of

equation (25): ΔS/S= (r – σ2/2)Δt+ σdZ= (r – σ2/2)Δt+ σ tΔε . Since ε is distributed

like a standard normal random variable, ΔS/S~N((r – σ2/2)Δt, σ tΔ ).

Each week, use the following steps to determine the simulated stock price:

(i) Set i=0.

(ii) Generate a random value v1 from a standard normal distribution. (Standard

spreadsheet tools include this capability.)

(iii) Convert v1 to a sample v2 from a N((r – σ2/2)Δt, σ tΔ ) by setting v2 = (r –

σ2/2)Δt + σ tΔ v1.

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(iv) Set ΔS=v2S. ΔS represents the incremental change in stock price from the prior

period to the current period.

(v) , where is the simulated updated value of the stock price after one

period.

SSS Δ+=′ S ′

(vi) Set , i=i+1. SS ′=

(vii) If i=T then stop. S is the simulated stock price at the end of six months. If i<T,

return to step (i).

Note that randomness only enters in step (ii) when generating a random value v1. All

other steps are mere transformations or calculations and are deterministic. The payoff of

a call option at expiration is given by equation (21). Further, in the absence of arbitrage

opportunities (i.e., assuming efficient markets) and by applying the theory of risk neutral

valuation we know that the value of the option is equal to its expected payoff discounted

by the risk free rate. Using these facts, we apply the Monte Carlo simulation method to

price the option. The overall methodology is as follows:

(i) Define the number of time periods until maturity, T.

(ii) Use Monte Carlo simulation to simulate a path of length T describing the

evolution of the underlying stock price, as described above. Denote the final

stock price at the end of this simulation by SF.

(iii) Determine the option payoff according to equation (21), assuming SF, the final

period T stock price determined in step (ii).

(iv) Discount the option payoff from step (iii) assuming the risk-free rate. The

resulting value is the current value of the option.

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(v) Repeat steps (ii)-(iv) until the confidence bound on the estimated value of the

option is within an acceptable range.

10.4. A numerical example

A stock has expected annual return of μ=15% per year and standard deviation of

σ=20% per year. The current stock price is S=$42. An investor wishes to determine the

value of a European call option with a strike price of $40 that matures in six months. The

risk free rate is 8%.

We will use Monte Carlo simulation to simulate the path followed by the stock price

and hence the stock price at expiration which determines the option payoff. We consider

weekly time intervals, i.e., Δt=1/52. Thus T=24 assuming, for the sake of simplicity, that

there are four weeks in each month.

To simulate the path of the stock price over the 24 week period, we follow the

algorithm described in Section 10.3. We first compute the risk-neutral drift (r-σ2/2)Δt,

which with these parameter settings works out to be 0.0012. The random quantity ε is

distributed like a standard normal random variable, so ΔS/S~N(.0012,.0277).

The starting price of the stock is $42. Each week, use the following steps to

determine the simulated updated stock price:

(i) Generate a random value v1 from a standard normal distribution. (Standard

methods can be used.)

(ii) Convert v1 to a sample v2 from a N(.0012,.0277) by setting v2 = .0012+.0277v1.

(iii) Set ΔS=v2S.

(iv) Set SSS Δ+=

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Steps (i)-(iv) yield the simulated updated stock price after one week. Repeat this

process T=24 times to determine SF, the stock price at the end of six months. Then, the

option payoff equals P=max(SF-40,0). P is the option payoff based upon a single

simulation of the ending stock price after six months, i.e., based upon a single simulation

run. Perform many simulation runs and after each run compute the arithmetic average

and confidence bounds of the simulated values of P. When simulation runs have been

performed so that the confidence bounds are acceptable, the value of the option can be

computed based upon the expected value of P: V= . )()5(.08. PEe−

11. Dynamic programming

Dynamic programming is formal method for performing optimization over time.

The algorithm involves breaking a problem into a number of subproblems, solving the

smaller subproblems, and then using those solutions to help solve the larger problem.

Similar to stochastic programming with recourse, dynamic programming involves

sequential decision making where decisions are made, information is revealed, and then

new decisions are made. More formally, the dynamic programming approach solves a

problem in stages. Each stage is comprised of a number of possible states. The optimal

solution is given in the form of a policy which defines the optimal action for each stage.

Taking action causes the system to transition from one stage to a new state in the next

stage.

There are two types of dynamic programming settings: deterministic and

stochastic. In a deterministic setting, there is no system uncertainty. Given the current

state and the action taken, the future state is known with certainty. In a stochastic setting,

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taking action will select the probability distribution for the next state. For the remainder

we restrict our discussion to a stochastic dynamic programming setting, since finance

problems are generally not deterministic. If the current state is the value of a portfolio,

and possible actions are allocations to different assets, the value of the portfolio in the

next stage is not known with certainty (assuming that some of the assets under

consideration contain risk).

A dynamic program typically has the following features:

(i) The problem is divided into t=1,…,T stages. xt denotes the state at the

beginning of stage t and at(xt) denotes the action taken during stage t given

state xt. Taking action transitions the system to a new state in the next

stage so that xt+1=f(xt, at(xt), εt), where εt is a random noise term. The

initial state x0 is known.

(ii) The cost (or profit) function in period t is given by gt(xt, at(xt), εt). This

cost function is additive in the sense that the total cost (or profit) over the

entire T stages is given by:

∑ −

=+

1

1)),(,()),(,( T

t tttttTTTTT xaxgxaxg εε . (26)

The objective is to optimize the expected value of equation (26).

(iii) Given the current state, the optimal solution for the remaining states is

independent of any previous decisions or states. The optimal solution can

be found by backward recursion. Namely, the optimal solution is found

for the period T subproblem, then for the periods T-1 and T subproblem,

etc. The final period T subproblem must be solvable.

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The features dynamic program that defines the options pricing problem differ

somewhat from the features described here. In Section 12.1 we note these differences.

12. Pricing American options using dynamic

programming

Monte Carlo simulation is a powerful tool for options pricing. It performs well

even in the presence of a large number of underlying stochastic factors. However, at

each step simulation progresses forward in time. On the other hand, options that allow for

early exercise must be evaluated backward in time where in each period the option holder

must compare the intrinsic value of the option against the expected future value of the

option.

The traditional approach to pricing American options has been to use binomial

trees and dynamic programming. Dealing with early exercise requires one to go

backwards in time, as at each decision point the option holder must compare the value of

exercising immediately against the value of holding the option. The value of holding the

option is simply the price of the option at that point.

In this section we will show how one can use dynamic programming to price an

American option. The method involves two steps:

(i) Build a T stage tree of possible states. The states correspond to points

visited by the underlying stock price process.

(ii) Use dynamic programming and backward recursion to determine the

current value of the option.

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12.1. Building the T stage tree

Cox, Ross, and Rubinstein, (1979) derived an exact options pricing formula under

a discrete time setting. Following their analysis, we model stock price as following a

multiplicative binomial distribution: if the stock price at the beginning of stage t is S then

at the beginning of stage t+1 the stock price will be either uS or dS with probability q and

(1-q), respectively. Each stage has length Δt. We will build a “tree” of possible stock

prices in any stage. When there is only one stage remaining, the tree looks like:

uS with probability q

S

dS with probability 1 – q

Suppose there are two stages. In each stage, the stock price will move by “up” by

a factor of u with probability q and “down” by a factor of d with probability (1-q). In this

case, there are three possible final stock prices and the tree is given by:

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u2S with probability q2

uS

S duS=S with probability 2q(1-q)

dS

d2S with probability (1-q)2

The tree continues to grow according to this method. In general, at stage t there

are t+1 possible stock prices (states) that will appear on the tree. These are given by:

ujdt-jS, for j=0,…,t.

The values of u, d, and q are determined based upon the assumptions of efficient

markets, risk neutral valuation, and the fact that the variance of the change in stock price

is given by (from Section tΔ2σ 10.3). These values are:

dudaq

edeu

t

t

−−

=

==

Δ−

Δ

σ

σ

where . trea Δ=

12.1. Pricing the option using the Binomial Tree

We now use backward enumeration through the binomial tree to determine the

current stage 0 value of the option. We will illustrate the concept using the trees

developed in Section 12.1 above. Let K denote the strike price. With one period

remaining, the binomial tree had the form:

uS with probability q

S

dS with probability 1 – q

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The corresponding values of the call at the terminal points of the tree are

Cu=max(0,uS-K) with probability q and Cd=max(0,dS-K) with probability (1-q). The

current value of the call is given by the present value (using the risk free rate) of qCu+(1-

q)Cd.

When there is more than one period remaining, each node in the tree must be

evaluated by comparing the intrinsic value of the option against its continuation value.

The intrinsic value is the value of the option if it is exercised immediately; this value is

determined by comparing the current stock price to the option strike price. The

continuation value is the discounted value of the expected cash payout of the option

under the risk neutral measure, assuming that the optimal exercise policy is followed in

the future. Thus, the decision is given by:

]|)([),,0max(max 11 ttttr

tt xxgeEKxg ++Δ−−= . (27)

The expectation is taken over the risk neutral probability measure. xt, the current state, is

the current stock price. Notice that the action taken in the current state (whether or not to

exercise) does not affect the future stock price. Further, this value function is not additive.

However, its recursive nature makes a dynamic programming solution method useful.

12.2. A numerical example

We illustrate the approach using an identical setting as that used to illustrate the

Monte Carlo simulation approach to options pricing. However, here we will assume that

that we are pricing an American option. The investor wishes to determine the value of an

American call option with a strike price of $40 that matures in one month. (We consider

only one month to limit the size of the tree that we build.) The current stock price is

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S=$42. The stock return has standard deviation of σ=20% per year. The risk free rate is

8%.

We first build the binomial tree and then use the tree to determine the current

value of the option. We consider weekly time intervals, i.e., Δt=1/52. Thus T=4

assuming, for the sake of simplicity, that there are four weeks in each month.

493.0

973.0028.1

=−−

=

====

Δ

Δ−

Δ

dudeq

edeu

tr

t

t

σ

σ

The tree of stock movements over the four week period looks like:

u4S u3S u2S u2S uS uS S S S dS dS d2S d2S

d3S d4S We will evaluate each node in the tree by backward evaluation starting at the fourth time

period and moving backward in time. For each node we will use equation (write the

equation number) to compare the intrinsic value of the option against its continuation

value to determine the value of that node. The binomial tree for the value of the

American call option is:

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(6.92,0) (5.64,5.63) (4.39,4.39) (4.39,0) (3.18,3.21) (3.18,3.17) (2,2.28) (2,2.068) (2,0) (.85,1.26) (.85,.98) (0,.485) (0,0)

(0,0) (0,0) Every node in the tree contains two numbers in parenthesis. The first number is the

intrinsic value of the option. The second number is the discounted expected continuation

value, assuming that optimal action is followed in future time periods. The option value

at time zero (current time) is 2.28. Note that although the option is currently in the money,

it is not optimal to exercise even though it is an American option and early exercise is

allowed. By using the binomial tree to evaluate the option we find that the expected

continuation value of the option is higher than its current exercise value.

13. Comparison of Monte Carlo simulation and

Dynamic Programming

Dynamic programming is a powerful tool that can be used for pricing options

with early exercise features. However, dynamic pricing suffers from the so-called curse

of dimensionality. As the number of underlying variables increases the time required to

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solve the problem grows significantly. This reduces the practical use of dynamic

programming as a solution methodology. The performance of Monte Carlo simulation is

better in the sense that its convergence is independent of the state dimension. On the

other hand, as we have discussed, simulation has traditionally been viewed as

inappropriate for pricing options with early exercise decisions since these require

estimates of future values of the option and simulation only moves forward in time.

However, recent research has focused on combining simulation and dynamic

programming approaches to pricing American options to gain the benefits of both

techniques. See, for example, Broadie and Glasserman (1997).

14. Multi-period Asset Liability Management

The management of liability portfolios of relatively long-term products, like

pensions, variable annuities, and some insurance products requires a perspective that

goes beyond a single investment period. The portfolio optimization models of Sections

5 through 7 are short-term models. Simply rolling these models over into the next time

horizon can lead to problems. First, the models may make an excessive number of

transactions. Transactions are not free, and realistic portfolio management models must

take trading costs into consideration. Second, the models depend only on second

moments. Large market moves, such as during a market crash, are not part of the

model assumptions. Finally, policies and regulations place conditions on the

composition of portfolios. These are not part of the model assumptions.

Academic and finance industry researchers have, over the past few decades, been

exploring the viability of using multi-period balance sheet modeling to address the

issues of long-dated asset liability management. A multi-period balance sheet model

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treats the assets and liabilities as generating highly aggregated cash flows over multiple

time periods. The aggregations are across asset classes, so that individual securities in

an asset class, say stocks, are accumulated into a single asset class, say S&P 500. Other

asset classes are corporate bonds of various maturity classes, and so forth. The asset

class cash flows are aggregated over time periods, typically three or six months, so that

cash flows occurring within a time interval, say, ],1( tt − , are treated as if they all occur

at the end-point t. The model treats the aggregate positions in each asset category as

variables in the model. There is a single decision to be made for each asset class at the

beginning of each time period, which is the change in the holdings of each asset class.

The asset holdings and liabilities generate cash flows, which then flow into account

balances. These account balances are assigned targets, and the objective function

records the deviation from the targets. The objective of the model is to maximize the

sum of the expected net profits and the expected penalties for missing the targets over

the time horizon of the model.

A simplified application of such a model to a property-casualty insurance problem

is as follows. Let denote a vector of asset class values at time t and denote their

cash flows (e.g., interest payment, dividends, etc.) at time Let denote the portfolio

of holdings in the asset classes. Cash flows are generated by the holdings and by asset

sales:

tA ti

.t tx

1−+Δ= ttttt xixAC

where . The cash flows are subject to market and economic variability

over the time horizons of interest, say

1: −−=Δ ttt xxx

.,,1 Tt K=

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Liability flows from the property-casualty portfolio are modeled by aggregating

and then forecasting cash flows. The net liability flows are losses minus premium

income. Loss events are classified by frequency of occurrence and intensity given loss.

These can be simulated over the time horizon T using actuarial models for insurance

payments. The evolution of the liability portfolio composition (say, by new sales, lapses

of coverage, and so forth) can also be modeled. The key correlation to capture in the

liability model is the relationship between the state of economic activity and the asset

markets. For example, high employment is indicative of strong economic activity, which

can lead to increases in the insurance portfolio; high inflation will lead to higher loss

payouts given losses; and so forth.

tL

Various performance, accounting, tax, and regulatory measurements are

computed from the net cash flows. For example, one measurement could be

shareholder’s equity at the time horizon , another could be annual net income , and

yet another could be premium-surplus ratio – a quantity used in the property-casualty

industry as a proxy for the rating of the insurance company.

TS tN

tP

In these aggregated models, we try to model the change in performance

measurements as linear computations from the cash and liability flows and the previous

period quantities. For example, net income is computed as cash flow minus operating

expenses. If is a proxy for the contribution of portfolio management to expenses,

for example the cost of trading, then net income can be modeled by the following

equation

tt xO Δ

. ttttt xOLCN Δ−−=

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Shareholder’s equity can change due to a number of influences; here we just capture the

change due to the addition of net income

ttt NSS += −1 .

Finally, premium-surplus ratio can be approximated by fixing premium income to a level

L and assuming (this is a major simplification!) that the surplus is equivalent to

shareholders equity:

tt SLP /= .

A typical objective for a multi-period balance sheet model/an asset-liability matching

problem is to create performance targets for each quantity and to penalize the shortfall.

Suppose that the targets are tN for annual net income, TS for shareholder’s equity,

and tP for premium-surplus ratio. Then the objective function could be

Maximize ][][ ∑∑ ++ −−−− ttt

ttT PSLNNSE μλ

Subject to 1

1

−

−

+=Δ−−+Δ=

ttt

ttttittt

SNSxOLxixAN

(28)

where the parameters λ and μ are used to balance the various contributions in the

objective function, the premium-surplus ratio target relationship has been multiplied

through by the denominator to make the objective linear in the decision variables, and

the objective is integrated over the probability space represented by the discrete

scenarios.

The objective function in formulation (28) can be viewed as a variation of the

Markowitz style, where we are modeling “expected return” through the shareholder’s

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equity at the end of the horizon, and “risks” through the shortfall penalties relative to

the targets for net income and premium-surplus ratio.

14.1. Scenarios

In multi-period asset liability management the probability distribution is modeled

by discrete scenarios. These scenarios indicate the values, or states, taken by the random

quantities at each period in time. The scenarios can branch so that conditional

distributions given a future state can be modeled. The resulting structure is called a

“scenario tree’’. Typically there is no recombining of states in a scenario tree, so the size

of the tree grows exponentially in the number of time periods. For example, in the

property-casualty model, the scenarios are the values and cash flows of the assets and the

cash flows of the liabilities. The values of these quantities a each time point t and

scenario s is represented by the triple . The pair is sometimes called a

“node” of the scenario tree. The scenario tree may branch at this node, in which case the

conditional distribution for the triple given node is the values of the

triples on the nodes that branch from this node.

),,( st

st

st LiA ),( ts

),,( 111 +++ ttt LiA ),( ts

It is important to model the correlation between the asset values and returns and

the liability cash flows in these conditional distributions. Without the correlations, the

model will not be able to find positions in the asset classes that hedge the variability of

the liabilities. In property-casualty insurance, for example, it is common to correlate the

returns of the S&P 500 and bonds with inflation and economic activity. These

correlations can be obtained from historical scenarios, and conditioned on views as

discussed above in Section 7.

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The scenario modeling framework allows users to explicitly model the probability

and intensity of extreme market movements and events from the liability side. One can

also incorporate “market crash” scenarios in which the historical correlations are changed

for some length of time that reflects unusual market or economic circumstances – such as

a stock market crash or a recession. Finally, in these models it is usual to incorporate the

loss event scenarios explicitly rather than follow standard financial valuation

methodology, which would tend to analyze the expected value of loss distributions

conditional on financial return variability. Such methodology would ignore the year-to-

year impact of loss distribution variability on net income and shareholder’s equity.

However, from the ALM perspective, the variability of the liability cash flows is very

important for understanding the impact of the hedging program on the viability of the

firm.

14.2. Multi-period Stochastic programming

The technology employed in solving an asset liability management problem such

as this is multiperiod stochastic linear programming. For a recent survey of stochastic

programming, see Shapiro and Ruszczynski (2003).

The computational intensity for these models increases exponentially in the

number of time periods, so the models must be highly aggregated and strategic in their

recommendations. Nevertheless, the models do perform reasonably well in practice,

usually generating 300 basis points of excess return over the myopic formulations

based on the repetitive application of one period formulations, primarily through

controlling transaction costs and because the solution can be made more robust by

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explicitly modeling market crash scenarios. A recent collection of this activity is in the

volume edited by Ziemba and Mulvey (1998). See also Ziemba’s monograph Ziemba

(2003) for an excellent survey of issues in asset liability management.

15. Conclusions

In this chapter we saw the profound influence of applications of Operations

Research to the area of finance and financial engineering. Portfolio optimization by

investors, Monte Carlo simulation for risk management, options pricing, and asset

liability management, are all techniques that originated in Operations Research and found

deep application in finance. Even the foundations of options pricing are based on deep

applications of duality theory. As the name financial engineering suggests, there is a

growing part of the body of financial practice that is regarded as a subdiscipline of

engineering in which techniques of applied mathematics and operations research are

applied to the understanding of the behavior and the management of the financial

portfolios underpinning critical parts of our economy in capital formation, economic

growth, insurance, and economic-environmental management.

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